Climate change is the byproduct of the desired outcome, energy.
Advanced AI, if you buy Yuval's argument is the threat in and of itself.
So Climate Change is a problem that can be 'solved' while the main goal is pursued. This is ideologically consistent with gates investment in terrapower. Where as AI isn't because the desired outcome is the threat not a bi-product.
So your questions a bit flawed fundamentally.
As for gate's point is it true, almost certainly yes, the game theory is peruse and lie that you aren't, or openly pursue. You can't ever not pursue because you do not and cannot have perfect information.
Imagine how much visibility china would demand from the US to trust it was doing nothing, far more than they could give, and vice versa.
Do you think the us is going to give its adversaries tracking and production information its most advanced chips? It would never, and if they did why would other powers trust it if theres every reason to lie.